Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions

Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system's complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Quantum dynamics simulations of the 2D spectroscopy for exciton polaritons

We develop an accurate and numerically efficient non-adiabatic path-integral approach to simulate the non-linear spectroscopy of exciton-polariton systems. This approach is based on the partial linearized density matrix approach to model the exciton dynamics with explicit propagation of the phonon bath environment, combined with a stochastic Lindblad dynamics approach to model the cavity loss dynamics. Through simulating both linear and polariton two-dimensional electronic spectra, we systematically investigate how light-matter coupling strength and cavity loss rate influence the optical response signal. Our results confirm the polaron decoupling effect, which is the reduced exciton-phonon coupling among polariton states due to the strong light-matter interactions. We further demonstrate that the polariton coherence time can be significantly prolonged compared to the electronic coherence outside the cavity.

Equivalence of quantum and classical third order response for weakly anharmonic coupled oscillators

Two-dimensional (2D) infrared (IR) spectra are commonly interpreted using a quantum diagrammatic expansion that describes the changes to the density matrix of quantum systems in response to light-matter interactions. Although classical response functions (based on Newtonian dynamics) have shown promise in computational 2D IR modeling studies, a simple diagrammatic description has so far been lacking. Recently, we introduced a diagrammatic representation for the 2D IR response functions of a single, weakly anharmonic oscillator and showed that the classical and quantum 2D IR response functions for this system are identical. Here, we extend this result to systems with an arbitrary number of bilinearly coupled, weakly anharmonic oscillators. As in the single-oscillator case, quantum and classical response functions are found to be identical in the weakly anharmonic limit or, in experimental terms, when the anharmonicity is small relative to the optical linewidth. The final form of the weakly anharmonic response function is surprisingly simple and offers potential computational advantages for application to large, multi-oscillator systems.

An accurate and efficient Ehrenfest dynamics approach for calculating linear and nonlinear electronic spectra

Linear and nonlinear electronic spectra provide an important tool to probe the absorption and transfer of electronic energy. Here, we introduce a pure state Ehrenfest approach to obtain accurate linear and nonlinear spectra that is applicable to systems with large numbers of excited states and complex chemical environments. We achieve this by representing the initial conditions as sums of pure states and unfolding multi-time correlation functions into the Schrödinger picture. By doing this, we show that one can obtain significant improvements in accuracy over the previously used projected Ehrenfest approach and that these benefits are particularly pronounced in cases where the initial condition is a coherence between excited states. While such initial conditions do not arise when calculating linear electronic spectra, they play a vital role in capturing multidimensional spectroscopies. We demonstrate the performance of our method by showing that it is able to quantitatively capture the exact linear, 2D electronic spectroscopy, and pump-probe spectra for a Frenkel exciton model in slow bath regimes and is even able to reproduce the main spectral features in fast bath regimes.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

A Semiclassical Framework for Mixed Quantum Classical Dynamics

Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.